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Question

The vector $$\overrightarrow{c}$$ directed along the internal bisector of the angle between the vectors $$\overrightarrow{a}=2\hat{i}-\hat{j}-2\hat{k}$$ and $$\overrightarrow{b} =2\hat{i}+2\hat{j}+\hat{k}$$ with $$\left|\overrightarrow{c}\right|=3\sqrt{2}$$ is


A
4^i^j^k
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B
4^i^j+^k
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C
4^i+^j^k
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D
4^i+^j+^k
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Solution

The correct option is D $$4\hat{i}+\hat{j}-\hat{k}$$
The vector along internal bisectors of angle between $$\overrightarrow{a}$$ and $$\overrightarrow{b}$$ is along $$\dfrac{\overrightarrow{a}}{\left|\overrightarrow{a}\right|}+\dfrac{\overrightarrow{b}}{\left|\overrightarrow{b}\right|}$$
$$\therefore \dfrac{2\hat{i}-\hat{j}-2\hat{k}}{\left|2\hat{i}-\hat{j}-2\hat{k}\right|}+\dfrac{2\hat{i}+2\hat{j}+\hat{k}}{\left|2\hat{i}+2\hat{j}+\hat{k}\right|}$$
$$ =\dfrac{4\hat{i}+\hat{j}-\hat{k}}{3}$$ on simplification 
$$\therefore \left|2\hat{i}-\hat{j}-2\hat{k}\right| = \sqrt{4+1+4}=3$$
$$\Rightarrow \left|\overrightarrow{c}\right|=3\sqrt{2}$$(given)
$$\Rightarrow \overrightarrow{c}=4\hat{i}+\hat{j}-\hat{k}$$

Mathematics

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